Question: Factor the following expression: $7$ $x^2+$ $24$ $x$ $-16$
Answer: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(7)}{(-16)} &=& -112 \\ {a} + {b} &=& & & {24} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-112$ and add them together. Remember, since $-112$ is negative, one of the factors must be negative. The factors that add up to ${24}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-4}$ and ${b}$ is ${28}$ $ \begin{eqnarray} {ab} &=& ({-4})({28}) &=& -112 \\ {a} + {b} &=& {-4} + {28} &=& 24 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {7}x^2 {-4}x +{28}x {-16} $ Group the terms so that there is a common factor in each group: $ ({7}x^2 {-4}x) + ({28}x {-16}) $ Factor out the common factors: $ x(7x - 4) + 4(7x - 4) $ Notice how $(7x - 4)$ has become a common factor. Factor this out to find the answer. $(7x - 4)(x + 4)$